help with these questions~
Thank you
1. Determine the interval(s) on which x^2 + 2x -3>0
a)x<-3, x>1
b)-3<x<1
c)x<-3, -3<x<1, x>1
d) x>1
Answer is D, x>1
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2. Determine when the function f(x)= 3x^3 + 4x^2 -59x -13 is greater than 7.
a) -5<x<1/3
b) -5<x<-1/3, x>4
c) x<4
d) x=-5, -1/3,4
Answer is: B
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3. Provde the intervals you would check to determine when -5x^2 + 37x> -15x^2 + 12x + 15.
a) x=3, x=-0.5
b) x<-3, -3<x<0.5, x>0.5
c) x<-1, -1<x<15, x>15
d) x= 1, x=15
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To solve these questions, we need to use algebraic manipulation and graphing techniques.
1. To determine the interval(s) on which x^2 + 2x - 3 > 0, we can start by factoring the quadratic equation (x^2 + 2x - 3) into (x - 1)(x + 3). Now we have (x - 1)(x + 3) > 0. To find when this quadratic equation is greater than zero, we need to determine the sign of the expression (x - 1)(x + 3) for different intervals of x.
To do this, we can create a sign chart by selecting test points in each interval.
Let's choose x = 0 as our test point.
For (x - 1)(x + 3) > 0,
if x < -3, both factors (x - 1) and (x + 3) are negative, so the expression is positive.
if -3 < x < 1, (x - 1) is negative and (x + 3) is positive, so the expression is negative.
if x > 1, both factors (x - 1) and (x + 3) are positive, so the expression is positive.
Based on the sign chart, we can see that the expression (x - 1)(x + 3) is positive when x > 1. Therefore, the correct answer is d) x > 1.
2. To determine when the function f(x) = 3x^3 + 4x^2 - 59x - 13 is greater than 7, we set f(x) greater than 7:
3x^3 + 4x^2 - 59x - 13 > 7.
Now we can rearrange the equation to get:
3x^3 + 4x^2 - 59x - 20 > 0.
To find the intervals where this cubic function is greater than zero, we can use a similar approach as before and create a sign chart using test points in different intervals.
Choosing x = 0 as a test point:
For 3x^3 + 4x^2 - 59x - 20 > 0,
if x < -5, the expression is negative.
if -5 < x < -1/3, the expression is positive.
if x > 4, the expression is positive.
Based on the sign chart, we can see that the expression 3x^3 + 4x^2 - 59x - 20 is positive when -5 < x < -1/3 and x > 4. Therefore, the correct answer is b) -5 < x < -1/3, x > 4.
3. To determine the intervals where -5x^2 + 37x > -15x^2 + 12x + 15, we can start by simplifying the equation:
-5x^2 + 37x > -15x^2 + 12x + 15.
By rearranging terms, we get:
10x^2 - 25x - 15 > 0.
To find the intervals where this quadratic inequality is greater than zero, we can use either factoring or the quadratic formula to find the x-intercepts of the quadratic equation. However, in this case, it is easier to complete the square:
10(x^2 - 2.5x - 1.5) > 0.
Now we complete the square in the brackets:
10(x^2 - 2.5x + 1.25 - 1.25 - 1.5) > 0.
This simplifies to:
10((x - 1.25)^2 - 2.75) > 0.
Now we can factor out the 10:
10(x - 1.25)^2 - 27.5 > 0.
Dividing both sides by 10:
(x - 1.25)^2 - 2.75 > 0.
We can see that a perfect square is always greater than or equal to zero, so we know that the expression (x - 1.25)^2 is always greater than or equal to 0, and it is equal to 0 when x = 1.25.
Therefore, for the inequality to be true, we need the expression (x - 1.25)^2 - 2.75 to be greater than 0. This means that x needs to be either less than 1.25 - √2.75 or greater than 1.25 + √2.75.
Calculating the square root of 2.75, we get approximately √2.75 ≈ 1.6583.
Hence, the correct answer is c) x < -1.25 - √2.75, -1.25 + √2.75 < x < -1.25 + √2.75, x > -1.25 + √2.75.